The resistance cube.To simplify given circuits is an art of electrical engineers. Often it turns out that a complicated problem seen from a different point of view is extremely easy. The resistance cube ("R-cube") is one such example. The solution we will find can also be used to calculate the impedances of more complicated elements like the "C-cube" or the oscillatory "RLC-cube".

Problem Statement

Find the total resistance \(R_{c}\) of the resistance cube ("R-cube") shown below with respect to opposing edges. You may want to draw a two-dimensional circuit diagram first. There is a lot of redundancy in the cube since all resistances are equal to \(R\) - maybe you can find a way to simplify the circuit.

When you have successfully calculated the resistance of the “R-cube”, you can easily calculate the impedance \(Z_{c,C}\) of the “C-cube”, where all resistances are exchanged with equal capacitances \(C\). Assume some time-harmonic voltage source connected to the opposing edges. Can you also find the impedance \(Z_{c,RLC}\) of the “RLC-cube” as shown below with resistance \(R\), inducance \(L\) and capacitance \(C\)?

The resistance cube (R-cube), the C-cube and the RLC-cube

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